Recumbent Bicycle Design by Charles Meredith Brown - November 2015
Formula for Estimating the Air Drag
of Unfaired Recumbent Bicycles
"If I have seen farther than others, it is because I have copied off the test papers of giants."
   
Sections:
Wheel air drag
Handlebar air drag
Cycle component air drag
 
Seat angle vs air drag
Conclusions
Sources
Also see: Bicycle frame design, Steering and ride and Putting it all together


We have a wealth of data from people who have taken the trouble to measure the drag of recumbent bicycles in the wind tunnel and on the road! Why not pour over all the data, for if we put it all together we might devise a formula to predict the air drag of other bicycles! Prospective cycle buyers would have a better idea of what speed might be expected of various mounts. Inveterate tinkerers, forever coming up with more ideas for bikes than they have time to build, can winnow down the field and concentrate their energies on the more promising designs.

Wheel air drag

The only tests I could find anywhere comparing the air drag of different diameter wheels were those by Kyle. For regular non aero wheels, round spokes, and regular rims, air drag varies roughly with diameter. For disk and covered wheels, the advantage of smaller size is less, a 406 (20”) wheel having maybe 80% the air drag of a 700c. Aero spokes and rims fall in between the two.

Wheel covers can cut the air drag of a regular non- aero wheel in half. Since they are cutting down the wind resistance of the whirling spokes, they are effective even on the back wheel of a low racer, which is well shielded from the direct wind. On a typical unfaired recumbent, wheel covers front and back will increase speed about 3%.

Aero wheels actually make more of a difference than the chart indicates. In a crosswind, the air drag of standard wheels goes up, while that of disk and deep rim wheels goes down.

To quickly go over what just about everyone reading this already knows, the wheels with the lowest air drag are lenticular disks, where the sides curve out. In the right crosswind, the air drag of these wheels can drop below zero, the disks acting as sails to provide a propulsive force. Narrower tires and rims have less air drag. It is best to match up tire and rim widths, aero spokes and fewer spokes help, and aero rims cut drag.

People generally avoid using disk or covered wheels on the front, as they make the bike hard to control in a crosswind. However, as this varies not only with wheel size but with distance from the center of pressure to the steering axis, so a 406mm disk wheel would be affected only about 1/3 as much as a 700. Sensitivity to cross winds depends strongly on steering angle. With a shallow steering angle of 50 degrees, a disk wheel really gets blown around. However, some experimental remote steering bicycles that people have been building, with a steering axis around 90 degrees and a reversed fork, would be excellent geometry for a front disk wheel.

Most of the air drag of a wheel is towards the top. With no wind, the bottom of the wheel is not moving at all relative to the wind- thus zero air drag! Alas, the top of the wheel is moving along at twice the speed the bicycle is going. Since air drag goes up at the square of speed, this has four times the air drag.

On paper I tried dividing the surface of a wheel into small sections. I rotated the wheel just a tiny bit forward, and compared how much each bit moved, and the relative air drag of each bit. The top half of a wheel has to have over 2/3 of the air drag. If you have two 406 wheels with 28mm wide tires, with the standard non-aero construction, and add in the air drag of forks, rear derailleur, some chain and framework, this would have an air drag in CdA of about 0.06 to 0.07 square meters. In the course of putting together this article, I found that we weren’t as fast as we thought. Most of the early research on recumbent air drag compared us with diamond- frame bikes with the rider in a ’touring’ position- arms straight. As you all know, as soon as you challenge an upright rider to a race, they immediately drop to a crouch, reducing their air drag. “That’s cheating!” we cry, “Get back into the position we based our air drag calculations on!” Alas, all’s fair in love, war, and bicycle racing. Let’s not dwell upon the moral rectitude of a group that counts Lance Armstrong among their number, and instead base our drag comparisons on an upright rider in the race position.

Okay, let’s say you come across the local upright cycling club on a Sunday morning. They have just come out of church, where they sing soprano in the choir, and as they come closer they mercilessly deride your recumbent and expound loudly on how comfortable their seats are.

Do you:
1) Whine and make excuses for your bicycle’s poor performance?
2) Lead the group on a scenic tour of the local environs, stopping frequently to smell the flowers?
or:
3) Put your mettle to the pedals, blowing past the pack and leave them bobbling helplessly in your wake? Crushing their spirits with such sure certainty that they are left but quivering shells, with you upholding the honor of recumbentdom by dealing them the shattering comeuppance they so richly deserve?

In the off chance that any of my readers are brute enough to choose #3, or if you just like going fast, we had best continue to investigate ways to reduce air drag:

Air drag of handlebars

Under seat handlebars might not give such high wind resistance if the seat is laid way back, so the arms line up more with the air flow.

I have measured the air drag of some above seat steering handlebars, the test bike had a seat angle of 35 degrees and a bottom bracket 25cm (10") higher than the seat:

Handlebars behind the knees Handlebars beside the knees Handlebars ahead of knees

Other things being equal, the bike with handlebars behind the knees goes 3% faster than the other two. Care should be taken to keep the elbows from sticking out. Handlebars ahead of the knees would probably cause no drag increase on a bike with a lower bottom bracket.

Air drag of the mechanical doodads sticking out under the rider

Low racers get their air drag down so low by getting the rider into an aero position, then lowering the engine almost to the ground so the mechanical parts and rider are all drafting each other. They have little drag added by machinery hanging down below. In these ways air drag can be brought to half that of a diamond-frame bike with the rider in race position, and 20% faster for the same power.

The numbers below are my best estimates for air drag of parts exposed under the seat:

Front wheel, 406mm diameter, 28 mm wide, standard (non-aero) construction:
          

Rear 700c spoked, non-aero wheel with 28mm wide tire
Rear 700c disk wheel or with wheel covers, 28mm wide tire.

Note that on many recumbent bikes, the air drag of a 700 wheel can actually be a little less than that of a 406 wheel. Being taller, the upper, high-drag portion of a larger wheel is more likely to be blocked from the oncoming wind by the rider's body. Blocking this also blocks wind from hitting a lot of other mechanical parts.

Using the drag measurements from the sources listed above, then subtracting the air drag of the mechanical bits exposed to the wind, we get a pretty good idea of that of the rider. I tried different things, but seat back angle seemed to have the closest relationship to rider air drag.

In various tests, different size riders were used. In this chart, I've adjusted the results to a 90 kg. (198 lb.) total bike + rider weight, and assumed the drivetrain to be 95% efficient. Bike measurements were the best I could do working from photos, and will be a bit off.

Seat angle vs air drag

These are the assumptions I made in the "Air Drag vs. Seat Back Angle" graph, which is coming up:

Bike Rear
Wheel
Front
Wheel
Seat
Height (cm)
BB
Height
(cm)
Seat
Angle
(deg)
Rolling
Resistance
(newtons)
Total
CdA
(m2)
Mech
CdA
(m2)
Rider
CdA
(m2)
Data
Sources
Diamond frame, touring position 700 700 wrong bad snafu - 0.40 0.10 0.30 -
Diamond frame, racing crouch 700 700 fubar reject pooey - 0.32 0.10 0.22 -
Tour Easy (LWB) 700 406 53 33 72 - 0.31 0.07? 0.24 3,6,7
Vision Saber (highracer) 24" 24" 72 72 40 - 0.30 0.08 0.22 4
Lebsack High Racer 700 700 54 83 32 - 0.24 0.05 0.19 5
Flevobike 50/50 406 406 44 66 31 5.1 0.25 0.06 0.19 1
M-5 20/20 451 451 44 74 30 4.5 0.22 0.05 0.17 1
Affect 29̊ 451 451 56 78 29 4.5 0.26 0.07 0.19 2
Baron Low Racer 700 406 33 55 22 5.1 0.20 0.03 0.17 1
Affect 21̊ 451 451 56 78 21 4.5 0.24 0.07 0.17 2
M-5 Low-Racer 700 451 26 51 17 4.2 0.18 0.02 0.16 1

Comments on above:
Tour Easy- Source 3 says the Tour Easy has less drag than a diamond frame bike with the rider in a crouch position. Schondorf says a similar bike has higher drag, in my own experience bikes similar to the Tour Easy have similar drag to a fast upright. How much to allow for air drag of the mechanical parts? There's more guesswork in these numbers than any other bike here.
Vision Saber- A lot of hardware under the rider really brings up the air drag. Why didn't they make the seat lower?
M-5 20/20- unusually low air drag for this seat angle. The pedals are much higher than the seat, is this helping?
"Affect" bicycles are from source #2.

This graph is derived from the chart above:


The dotted line is figured as 0.13 x sine of angle plus 0.12. The line is chosen simply because it is a good fit to the information we currently have available. I think it is amazing how well these results agree considering the measurements were done by different people using very different methods.Clearly there is a trend these days towards more laid-back seats to reduce wind drag. The problem with such a design is bracing the rider so there is something to push against. Jay Hoover and I used to joke about strapping a rider into a Velcro vest which worked with a Velcro seat. Laid-back seats usually provide special support at the lumbar region and shoulders, and the rider learns to pull back with the lower foot to help keep from having their torso pushed around with each pedal stroke. Extrapolating from above, a rider flat on the back, legs flailing and head raised up a bit to see where you're going, would have a CdA of around 0.12.

Thanks to source #3 above, we actually have the air drag of a motionless person laying flat on their back. Scaled up to the size test rider we've been using, this works out to a CdA of 0.07. We can picture the scenario: the subject, described as a 109 lb. female student, was strapped to the wind tunnel scale, screaming, "No! No!" as Dr. Kyle, cackling maniacally, cranked up the fans.
And no one lipped off in Professor Kyle’s class ever again.

Conclusions

To minimize air drag on an unfaired recumbent
1) Use wheel covers or unfaired wheels
2) Get the rider into an aero position
3) Get the rider low, to block as much wind as possible from hitting the mechanicals.

So there's the formula for estimating air drag- take the rider's air drag, and add in the drag for each wheel and its attendant hardware. Allowing for rider size, I think it will get you within 10%. It's not perfect, but we need more R&D. Not included are more unusual positions, such as prones or the bent- forward Dopplers of Heisch and Schwartz. We HPV'ers are a creative lot, and soon I'm sure someone will do better.

Here's how to do the math:
Power required = drag x velocity (velocity in meters/second x drag in newtons = power in watts) (Aah, so much easier to use metric than the imperial system!)

Air drag = coefficient of drag x frontal area x q (dynamic pressure).
If the units are metric and standard air is used (by which aerodynamicists mean 15 degrees C. and sea level), q can be approximated by speed in meters per second squared divided by 1.632. Coefficient of drag times frontal area in square meters, variously abbreviated CdA, CwA, or CxA, times q, gives air drag force in newtons. It may sound like work now, but believe it or not, the math can be fun when you get into it!

Let's try a design exercise. We'll try to come up with the fastest possible unfaired recumbent. Of course you use very aero wheels, and lay the rider down flat if you can do so without losing power, achieving a CdA of 0.12 for the rider alone. We'll assume, as before, gross weight w/engine of 90 kg, driveline 95% efficient. Rider's seat is at hub height.

What size wheels will we use? From those wheel graphs a few pages ago, we guesstimate the air drag of the bottom halves of two wheels, rear derailleur and some chain at a CdA of 0.013 for two 406 wheels, 0.016 for 700's. Assuming extra-low rolling resistance racing tires of equivalent quality, we'll assign the 406 a Crr of 0.0044, the 700 we'll figure at 0.0030:

Drag at 30 kilometers per hour (18.64 mph)   Drag at 50 kilometers per hour (31.07 mph)
  700 wheels 406 (20") wheels     700 wheels 406 (20") wheels
Air drag 5.79 newtons 5.66 newtons   Air drag 16.08 newtons 15.72 newtons
Rolling drag 2.65 newtons 3.88 newtons   Rolling drag 2.65 newtons 3.88 newtons
Driveline losses 0.44 newtons 0.50 newtons   Driveline losses 0.99 newtons 1.03 newtons
Power required 74 watts 84 watts   Power required 274 watts 287 watts

So the smaller wheels save a little bit of air drag (remember, you are using the rider's body to block most of the wheel air drag), but the bigger wheels save a lot more power in rolling resistance. Use big wheels on this bike.

The moving bottom bracket, front wheel drive type of bike would allow the pieces to fit:



Obviously, I wasn't thinking too much about forward visibility when designing this, but I'm sure you can do better. I think we will be seeing more MBB's at races.

If you want the cranks attached to the frame so your feet don't swivel when you turn, a smaller front wheel might be the better choice. For some designs it might allow a lower bottom bracket, a lighter, stiffer frame, or make it easier to pass the drivetrain over it. All engineering is compromise!

The next (future) section of this report will be summing up what we've learned from the last three sections, applying it to more examples like the one above. I'm hoping you will be following along with the designs you're interested in. You should be able to learn a thing or two about various bikes before you even start to test ride, buy, or build them! Stay tuned!

To reach the author: CharlesMeredithBrown@gmail.com

Sources:

I would like to list my main sources, so that the inquisitive might dig into the subject as they desire. There are innumerable sources for wheel air drag, eventually I came to rely mostly on the old wind tunnel tests of spinning wheels done by Chester Kyle & co.:

"Aerodynamic Wheels" by Chester Kyle, "Bicycling" December 1985.
"Wind Tunnel Tests of Bicycle Wheels & Helmets" by Chester Kyle, "Cycling Science" March 1990.
"New Aero Wheel Tests" by Chester Kyle, "Cycling Science" March 1991.

For drag tests of the whole bike and rider, this report would have been impossible without the careful research of Bert Hoge and associates in the Netherlands:
1) “Power Requirements for Laid-Back Recumbents” by Bert Hoge and Jeroen Schasfort, “HPV Nieuws” no. 4, 1999. Dave Wilson got an English translation of the high points to the English-speaking world in “Human Power” No. 50, Spring 2000. Drag measurements are given of five recumbents, using an SRM power meter. This uses strain gauges on the cranks to measure power. With the “Dolphin”, the drag is inexplicably a little higher than it should be. A lot of things can slow you down- dragging brakes, suspension 'pogoing' etc. so I didn't use that one.

2) “Affect of the Posture and Rolling Resistance on the Required Effort to Ride a Recumbent” by Bert Hoge. Ligfiets& nr.3-2003 titled "DeMeetligfiets". English translation by Wartenhorst and Cantono in “HUFF”, the Australian HPV Newsletter, Vol. 7 issue 3, May-June 2004. In their tests, seat back angle seems to have more affect than it should. The first photo shows the test rider in what appears to be a light jacket against the 50F. cold, which could be the reason. Anyway, that’s why I didn’t include the higher seat angle tests.
3) “The Aerodynamics of Human-powered Land Vehicles” by Gross, Kyle, and Malewicki, “Scientific American”, December 1983.

Some of the graphs are reprinted in “New Unified Performance Graphs and Comparisons for Streamlined Human Powered Vehicles” by Douglas Malewicki in the “Second International Human Powered Vehicle Scientific Symposium” edited by Allan Abbott, IHPVA, 1984. Both articles are excellent reads, and the second serves as an excellent primer for the math involved, although alas in imperial units. The math is slightly simplified to concentrate on the main points, and driveline drag is neglected.

4) “Aerodynamic Performance of Vision Recumbents” by Grant Bower . www.visionrecumbentinfo.com In the wind tunnel Bower finds the Vision Saber, the first American production highracer, has about the same air drag as a diamond-frame bike with the rider in an aero position.

5) "Aerodynamic Drag of Standard and Recumbent Bicycles" by John Stout. "Human Power" Vol. 3 No. 4, Summer '85. Wind tunnel tests of an early highracer, built by front-wheel-drive pioneer Jon Lebsack. I used the wind drag measurements of the larger rider (weights not listed) simply because he looked lightly built in the photo!

6) "Requirements for an All-Weather Pedal-Driven Machine for Transport and Sport" by Paul Schondorf. The First Human Powered Vehicle Scientific Symposium, Allan Abbott, editor IHPVA. The results of the drag tests are much higher than other people report. The test rider is shown in a heavy coat, which is probably the reason.

7) My own tests- I find long-wheelbase recumbents- my own were much like the Tour Easy, but with the seat 4” lower- to have about the same drag as an upright with the rider in an aero crouch / aero bar position.
8) "Excessive Bicycle Tinkering and its Deleterious Effects on Relations with Females" by Prof. Archibald Crusty. Crusty, as usual, goes way off subject, but makes the pertinent observation that if you're not being paid to write something, you might as well have fun doing it.

9) "The Practical Cyclist" by Chip Haynes. A good gift for someone just getting into cycling, covers the basics. Its inclusion here is a feeble attempt to plug my friend's book.

The M5 website (www.m5-ligfietsen.nl) claims that with 250 watts of rider power output, its new carbon high racer will go 48.3 kilometers per hour, while the unfaired M5 low racer only trundles along at a pokey 43 km/h at the same power. This seems to be the source for the air drag rating for this same bike in the 'recumbents.com' HPV simulator notes. Do you see high racers blowing away the lowracers on the race tracks? I want more details on how they ran the tests.
  

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